Problem: Simplify the following expression: $ q = \dfrac{a - 7}{a - 1} - \dfrac{5}{7} $
Answer: In order to subtract expressions, they must have a common denominator. Multiply the first expression by $\dfrac{7}{7}$ $ \dfrac{a - 7}{a - 1} \times \dfrac{7}{7} = \dfrac{7a - 49}{7a - 7} $ Multiply the second expression by $\dfrac{a - 1}{a - 1}$ $ \dfrac{5}{7} \times \dfrac{a - 1}{a - 1} = \dfrac{5a - 5}{7a - 7} $ Therefore $ q = \dfrac{7a - 49}{7a - 7} - \dfrac{5a - 5}{7a - 7} $ Now the expressions have the same denominator we can simply subtract the numerators: $q = \dfrac{7a - 49 - (5a - 5) }{7a - 7} $ Distribute the negative sign: $q = \dfrac{7a - 49 - 5a + 5}{7a - 7}$ $q = \dfrac{2a - 44}{7a - 7}$